AI 中文总结
研究次数至多为d且雅可比行列式为1的多项式自同构集合\(A(n, d)\)，证明其为扎里斯基闭集，其不可约分支与雅可比猜想相关，分支维数至少为\(n^2 - 1\)，在特定条件下集合\(X(n,d)\)的一般元素是雅可比猜想反例。
AI 中文摘要
我们证明了次数至多为d且雅可比行列式Jac(F)=1的多项式自同构F: \(\mathbb{C}^n\to\mathbb{C}^n\) 的集合\(A(n, d)\)是扎里斯基闭集。特别地，雅可比行列式为1的多项式映射集合\(A(n,d)\)的每个不可约分支要么由多项式自同构构成，要么（一般地）由雅可比猜想的反例构成。而且每个这样的分支维数至少为\(n^2 - 1\)。特别地，若集合\(X(n,d)\)不可约，且\(n\geq3\)，\(d\geq6\)，则该集合的一般元素是雅可比猜想的反例。
英文摘要
We show that the set $A(n, d)$ of polynomial automorphisms $F : \Bbb C^n \to \Bbb C^n$ of degree at most $d$ and with $Jac(F ) = 1$ is Zariski closed. In particular every irreducible component of the set $A(n,d)$ of polynomial mappings with Jacobian $1$ is either composed with polynomial automorphisms or (generically) with counterexamples to the Jacobian Conjecture. Moreover every such component has dimension at least $n^2-1.$ In particular if the set $X(n,d)$ is irreducible, and $n\ge 3, d\ge 6$, then a generic element of this set is a counterexample to the Jacobian Conjecture.
